The exceptional locus in the Bertini irreducibility theorem for a morphism

We introduce a novel approach to Bertini irreducibility theorems over an arbitrary field, based on random hyperplane slicing over a finite field. Extending a result of Benoist, we prove that for a morphism $\phi \colon X \to $\mathbb{P}$^n$suchthat$X$ is geometrically irreducible and the nonempty fibers of $\varphi$ all have the same dimension, the locus of hyperplanes $H$ such that ${\varphi }^{-1}H$ is not geometrically irreducible has dimension at most $\mathrm{codim}\varphi (X)+1$. We give an application to monodromy groups above hyperplane sections.